Cohomology in the service of AQFT John E. Roberts Università di Roma II 1 / 19
Cohomology Cohomology is an important part of mathematics and so ubiquitous as to form part of essentially any mathematical theory. It comes in many varieties but there are also unifying aspects. 2 / 19
An Example Locally trivial G -bundles. F → B Exists open covering O i of B and isomorphisms φ i : F ↾ O i → O i × G . φ i φ − 1 ( x , g ) = ( x , z ij g ) , x ∈ O i ∩ O j . j z ij z jk = z ik , O i ∩ O j ∩ O k � = ∅ , 1-cocycle. Given z exists F z . ζ �→ F z not a 1-1 correspondence. Need equivalence relations. z ∼ z ′ if there is a y i ∈ G such that z ij y j = y i z ij if O i ∩ O j � = ∅ . Equivalence of G -bundles. 3 / 19
Typical Features Cohomology has a degree ∈ Z . In low degree cohomology classifies something: H 1 ( B , G ) classifies locally trivial G –bundles. Cohomology is the cohomology of some type of mathematical object, here the ˇ C ech ohomology of a topological space B . A cohomology has coefficients, here the group G . May provide examples, May help with general theory Cohomology classes can often be computed by cohomological methods. But cohomology may prove to be just an alternative language. For example the problem of the existence of a field algebra and a gauge group boils down to asking whether the 6-j symbols of the relevant tensor category which form a 3-cocycle are actually a 3-coboundary. There are then 3-j symbols which can be used to embed the category in the category of Hilbert spaces. 4 / 19
Superselection Theory Selection Criterion π ↾ O ⊥ ≃ π 0 ↾ O ⊥ , O ∈ K . A ∈ A ( O ⊥ ) . Exists unitary V = V O with V π ( A ) = π 0 ( A ) V , Identify π 0 ( A ) with A Set ρ ( A ) := V π ( A ) V ∗ . ρ endomorphism localized in O . Things might have gone differently. 5 / 19
Simplices of a Partially Ordered Set A 0-simplex of K is an element a ∈ K . A 1–simplex b of K consists of three elements ∂ 0 b , ∂ 1 b ⊂ | b | . A 2-simplex c of K consists of three 1–simplices ∂ 0 c , ∂ 1 c and ∂ 2 c with ∂ 0 ∂ 0 c = ∂ 0 ∂ 1 c , ∂ 1 ∂ 0 c = ∂ 0 ∂ 2 c and ∂ 1 ∂ 1 c = ∂ 1 ∂ 2 c together with a further element | c | , the support of c , such that ∂ i ∂ j c ⊂ | c | , for all i , j . Pick V a as above and set z ( b ) := V ∂ 0 b V ∗ ∂ 1 b then z ( ∂ 1 c ) = z ( ∂ 0 ) z ( ∂ 2 c ) so that z is a 1–cocycle. It follows from duality that z ( b ) ∈ A ( | b | ) . We have local coefficients as in sheaf cohomology. Net cohomology. Exist localized endomorphisms y ( a ) with z ( b ) ∈ ( y ( ∂ 1 b ) , y ( ∂ 0 b )) . 6 / 19
Solitons 1972 2-spacetime dimensions A ⊂ F z ∈ Z 1 ( A ) ⊂ Z 1 ( F ) . Exist two sets of localized endomorphisms y ℓ ( a ) and y r ( a ) with z ( b ) ∈ ( y ℓ ( ∂ 1 b ) , y ℓ ( ∂ 0 b )) and z ( b ) ∈ ( y r ( ∂ 1 b ) , y r ( ∂ 0 b )) . α -induction 7 / 19
Completeness of sectors 1980 Free field with gauge group G . Easy to see that there is a sector corresponding to each irreducible representation of G . Question of whether there are other sectors remained open for quite some time. a ) ∩ ∂ b ′ = ∂ b F ( O + | b ′ | ) = F ( O + ∂ 0 b ) ∨ F ( O + ∂ 1 b ) , b ∈ Σ 1 , b ) If ( O + ∂ 0 b ) ⊥ ( O + ∂ 1 b ) then F ( O + ∂ 0 b ) ∨ F ( O + ∂ 1 b ) is canonically isomorphic to F ( O + ∂ 0 b ) ⊗ F ( O + ∂ 1 b ) , b ∈ Σ 1 . Abstract conditions that can be verified in the case of the free field. Ciolli completeness for the Streater and Wilde model. 8 / 19
Essential Duality duality A d = A , where A d ( O ) = ∩{ A ( O 1 ) ′ : O 1 ⊥ O} . essential duality A dd = A d . Z 1 ( A ) ≃ Z 1 ( A dd ) . Wedge duality implies essential duality. The set of representations satisfying essential duality is closed under direct sums and subrepresentations. In the absence of duality a representation satisfying the selection criterion, i.e. an object of Rep ⊥ A , yields a cocycle in Z 1 t ( A d ) , the path-independent cocycles in Z 1 ( A d ) . 9 / 19
Curved Spacetime The advent of curved spacetime revitalized superselection theory. The obvious question being: how does the topology and causal structure of spacetime affect the superselection structure? Guido, Longo, J.E.R, Verch (2001) Σ ⊥ 1 = { b ∈ Σ 1 : ∂ 0 b ⊥ ∂ 1 b } has same number of connected components as in Minkowski space. No new solitonic phenomena. Theory of sectors goes through if set K of regular diamonds is directed. Standard use of cohomology. Interesting part of problem left open. 10 / 19
Homotopy Two notions of path, usual topological one and one starting from K , where a path is a concatenation of 1–simplices. We suppose that K is path-connected. Both notions of path lead to a notion of homotopy group. If K is directed, Σ ∗ admits a contracting homotopy. Let M be arcwise connected and Hausdorff and K a base for the topology of M consisting of arcwise and simply connected subsets of M , then π 1 ( M ) = π 1 ( K ) . (Ruzzi) z ∈ Z 1 ( A ) and p ∼ q then z ( p ) = z ( q ) . Set η z ([ p ]) := z ( p ) , [ p ] ∈ π 1 ( K , a 0 ) . Map from 1–cocycles equivalent in B ( H 0 ) to equivalent unitary representations of the homotopy group. 11 / 19
Diamonds The approach to superselection sectors in Guido et al was based on the notion of regular diamond. These have the disadvantage that their causal complements may not be pathwise connected. Ruzzi improved matters by taking K to be the set of diamonds. Given a spacelike Cauchy surface C we let G ( C ) denote the set of open subsets G of C of the form φ ( B ) for a chart ( U , φ ) of C where B is an open ball of R 3 with cl ( B ) ⊂ φ − 1 ( U ) . A diamond of M is then a subset O = D ( G ) where G ∈ G ( C ) for some spacelike Cauchy surface C . D ( G ) is the domain of dependence of G . K is a base for the topology of M . A diamond is an open, relatively compact, arcwise and simply connected subset. D ( G ) is a globally hyperbolic spacetime with spacelike Cauchy surface G . The causal complement of a diamond O ⊥ := {O 1 ∈ K : O 1 ⊥ O} is pathwise connected in K . 12 / 19
Causal Punctures Typically, K is not directed when Cauchy surfaces of M are compact. Cannot transport charge to or from infinity. Remove a point. The causal puncture of K at a point x ∈ M is K x := {O ∈ K : ( O − ) ⊥ x } . Can also think in terms of a subset of M M x = M\ X x = D ( C\{ x } ) for some spacelike Cauchy surface C containing { x } . Considered as a spacetime, M x is globally hyperbolic but an element O ∈ K x need not be a diamond of M x . Still K x is a basis for the topology of M x and, M x being arcwise connected, K x is pathwise connected. 13 / 19
Strategy First discuss superselection sectors for K x for all x and then ’glue‘ the results together to describe the superselection theory for K . The advantage of studying K x is that it behaves in much the same way as Minkowski space. Let A x denote the restriction of A to K x . Each A x must satsfy duality. If z ∈ Z 1 ( A ) is path-independent on K x for each x ∈ M , then z is path-independent on K . Hence the 1–cocycles of z ∈ Z 1 ( A ) are trivial in B ( H 0 ) for an arbitrary 4-dimensional globally hyperbolic spacetime. A set of cocycles, z x ∈ Z 1 t ( A x ) , x ∈ M , extends to Z 1 t ( A ) if and only if z x 1 ( b ) = z x 2 ( b ) whenever | b | ∈ K x 1 ∩ K x 2 . A similar result holds for arrows. 14 / 19
Endomorphisms Superselection theory comes alive when endomorphisms are introduced. K x is not necessarily directed and the definition of y z ( a ) is a variant on the traditional one. Given z ∈ Z 1 t ( A x ) , a ∈ Σ 0 ( K x ) and define y z O ( a )( A ) := z ( p ) Az ( p ) ∗ , A ∈ A ( O 1 ) , O 1 ⊥ O , where x ∈ O ∈ K , p is a path in K x with ∂ 1 p ⊂ O and ∂ 0 p = a . This definition does not depend on the chosen path and, letting O shrink to { x } , extends to an endomorphism of A ⊥ ( x ) , the C ∗ –algebra generated by the A ( O 1 ) with O 1 ∈ K x . z ( p ) y z ( ∂ 1 p )( A ) = y z ( ∂ 0 p )( A ) z ( p ) . y z ( a )( A ( a 1 )) ⊂ A ( a 1 ) for a 1 ∈ K x with a ⊂ a 1 . 15 / 19
Tensor Product Tensor product on Z 1 t ( A x ) : ( z ⊗ z 1 )( b ) := z ( b ) y z ( ∂ 1 b )( z 1 ( b )) , b ∈ Σ 1 ( K x ) , ( t ⊗ s ) a := t a y z ( a )( s a ) , a ∈ Σ 0 ( K x ) , where t ∈ ( z , z 1 ) , s ∈ ( z 2 , z 3 ) . The composition law ⊗ makes Z 1 t ( A x ) into a tensor C ∗ –category. 16 / 19
Conjugates Let z be a simple object of Z 1 t ( A x ) then a conjugate z of z is given by z ( b ) := y z − 1 ( ∂ 0 b )( z ( b ) ∗ ) , b ∈ Σ 1 ( K x ) . In a symmetric tensor C ∗ –category with ( ι, ι ) = C where every simple object has a conjugate every object with finite statistics has a conjugate. Let T be a symmetric tensor C ∗ –category with conjugates, subobjects and direct sums, each object having a statistical phase 1 then T is isomorphic to the symmetric tensor C ∗ –category of finite dimensional unitary representations of a compact group unique up to isomorphism. 17 / 19
Recommend
More recommend
